An important class of problems in Complexity Theory is called the Constraint Satisfaction problems (CSP), whose complexity is at worst NP-complete, and in certain cases can be in P. We shall review some results connecting the complexity of the CSP and the structural properties of a finite algebra corresponding to it. At the end of the lecture we shall briefly go over the latest results by G. Kun and another by Berman, Idziak, McKenzie, Valeriote and the speaker. The second seems to pinpoint a class of algebras for which $CSP$ will be in P.